Crystalline cohomology

Summary

Crystalline cohomology is a cohomology theory for varieties over a perfect field $k$ of characteristic $p > 0$. Its values are modules over the ring $W(k)$ of Witt vectors. Grothendieck proposed it and Berthelot developed it, to obtain a $p$-adic cohomology in characteristic $p$, where ℓ-adic cohomology with $\ell = p$ fails. When the variety lifts to $W(k)$, it is the algebraic de Rham cohomology of the lift. So it is a de Rham theory that does not depend on the choice of a lift. It carries a Frobenius action and is a main object of $p$-adic Hodge theory.